Complexity of Pure and Mixed Qubit Geodesic Paths on Curved Manifolds
- URL: http://arxiv.org/abs/2209.10661v2
- Date: Tue, 8 Nov 2022 22:01:30 GMT
- Title: Complexity of Pure and Mixed Qubit Geodesic Paths on Curved Manifolds
- Authors: Carlo Cafaro, Paul M. Alsing
- Abstract summary: We propose an information geometric theoretical construct to describe and understand the complex behavior of evolutions of quantum systems in pure and mixed states.
We analytically show that the evolution of mixed quantum states in the Bloch ball is more complex than the evolution of pure states on the Bloch sphere.
- Score: 0.0
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: It is known that mixed quantum states are highly entropic states of imperfect
knowledge (i.e., incomplete information) about a quantum system, while pure
quantum states are states of perfect knowledge (i.e., complete information)
with vanishing von Neumann entropy. In this paper, we propose an information
geometric theoretical construct to describe and, to a certain extent,
understand the complex behavior of evolutions of quantum systems in pure and
mixed states. The comparative analysis is probabilistic in nature, it uses a
complexity measure that relies on a temporal averaging procedure along with a
long-time limit, and is limited to analyzing expected geodesic evolutions on
the underlying manifolds. More specifically, we study the complexity of
geodesic paths on the manifolds of single-qubit pure and mixed quantum states
equipped with the Fubini-Study metric and the Sjoqvist metric, respectively. We
analytically show that the evolution of mixed quantum states in the Bloch ball
is more complex than the evolution of pure states on the Bloch sphere. We also
verify that the ranking based on our proposed measure of complexity, a quantity
that represents the asymptotic temporal behavior of an averaged volume of the
region explored on the manifold during the evolution of the systems, agrees
with the geodesic length-based ranking. Finally, focusing on geodesic lengths
and curvature properties in manifolds of mixed quantum states, we observed a
softening of the complexity on the Bures manifold compared to the Sjoqvist
manifold.
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